Algebra & Number Theory

The equations defining blowup algebras of height three Gorenstein ideals

Andrew Kustin, Claudia Polini, and Bernd Ulrich

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We find the defining equations of Rees rings of linearly presented height three Gorenstein ideals. To prove our main theorem we use local cohomology techniques to bound the maximum generator degree of the torsion submodule of symmetric powers in order to conclude that the defining equations of the Rees algebra and of the special fiber ring generate the same ideal in the symmetric algebra. We show that the ideal defining the special fiber ring is the unmixed part of the ideal generated by the maximal minors of a matrix of linear forms which is annihilated by a vector of indeterminates, and otherwise has maximal possible height. An important step in the proof is the calculation of the degree of the variety parametrized by the forms generating the height three Gorenstein ideal.

Article information

Algebra Number Theory, Volume 11, Number 7 (2017), 1489-1525.

Received: 19 June 2015
Revised: 17 October 2016
Accepted: 19 December 2016
First available in Project Euclid: 12 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 13A30: Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics
Secondary: 13D02: Syzygies, resolutions, complexes 13D45: Local cohomology [See also 14B15] 13H15: Multiplicity theory and related topics [See also 14C17] 14A10: Varieties and morphisms 14E05: Rational and birational maps

blowup algebra Castelnuovo–Mumford regularity degree of a variety Hilbert series ideal of linear type Jacobian dual local cohomology morphism multiplicity Rees ring residual intersection special fiber ring


Kustin, Andrew; Polini, Claudia; Ulrich, Bernd. The equations defining blowup algebras of height three Gorenstein ideals. Algebra Number Theory 11 (2017), no. 7, 1489--1525. doi:10.2140/ant.2017.11.1489.

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